with initial condition \(\Delta^{\alpha-1}x(t)|_{t=0}=x_{0}\), where \(0<\alpha<1 \) is a constant, \(\Delta^{\alpha}x\) is the Riemann-Liouville fractional difference operator of order α of x, and \(\mathbb{N}_{0}=\{0,1,2,\ldots\}\).

Keywords

oscillationforced fractional difference equationdamping term

MSC

26A3339A1239A21

1 Introduction

In the past few years, the theory of fractional differential equations and their applications have been investigated extensively. For example, see monographs [1–4]. In recent years, fractional difference equations, which are the discrete counterpart of the corresponding fractional differential equations, have comparably gained attention by some researchers. Many interesting results were established. For instance, see papers [5–20] and the references therein.

The oscillation theory as a part of the qualitative theory of differential equations and difference equations has been developed rapidly in the last decades, and there have been many results on the oscillatory behavior of integer-order differential equations and integer-order difference equations. In particular, we notice that the oscillation of fractional differential equations has been developed significantly in recent years. We refer the reader to [21–33] and the references therein. However, to the best of author’s knowledge, up to now, very little is known regarding the oscillatory behavior of fractional difference equations [18–20]. Unfortunately, the main results of paper [18] are incorrect. The main reason for the mistakes in [18] is an incorrect relation of \(t^{(\alpha-1)}\) and \(t^{(1-\alpha)}\). In fact, noting the definition of \(t^{(\alpha)}=\frac{\Gamma(t+1)}{\Gamma(t+1-\alpha)}\), it is easy to observe that \(t^{(\alpha-1)} t^{(1-\alpha)}\neq1\).

In this paper we investigate the oscillation of forced fractional difference equations with damping term of the form

with initial condition \(\Delta^{\alpha-1}x(t)|_{t=0}=x_{0}\), where \(0<\alpha<1 \) is a constant, \(\Delta^{\alpha}x\) is the Riemann-Liouville difference operator of order α of x, and \(\mathbb{N}_{0}=\{0,1,2,\ldots\}\).

Throughout this paper, we assume that

(A)

\(p(t)\) and \(g(t)\) are real sequences, \(p(t)>-1\), \(f:\mathbb{N}_{0}\times\mathbb{R}\rightarrow\mathbb{R}\), and \(xf(t,x)>0\) for \(x\neq0\), \(t\in\mathbb{N}_{0}\).

A solution \(x(t)\) of the Eq. (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory.

2 Preliminaries

In this section, we present some preliminary results of discrete fractional calculus.

Definition 2.1

where f is defined for \(s=a\ \operatorname{mod}(1)\), \(\Delta^{-\nu}f\) is defined for \(t=(a+\nu)\ \operatorname{mod}(1)\), and \(t^{(\nu)}=\frac{\Gamma(t+1)}{\Gamma(t+1-\nu)}\). The fractional sum \(\Delta^{-\nu}f\) maps functions defined on \(\mathbb{N}_{a} =\{a,a+1,a+2,\ldots\}\) to functions defined on \(\mathbb{N}_{a+\nu}= \{a+\nu,a+\nu+1,a+\nu+2,\ldots\}\), where Γ is the gamma function.

Proof

Suppose to the contrary that there is a nonoscillatory solution \(x(t)\) of Eq. (1) which has no zero in \(\mathbb{N}_{t_{0}}=\{ t_{0},t_{0}+1,t_{0}+2,\ldots\}\). Then \(x(t)>0\) or \(x(t)<0\) for \(t\in\mathbb {N}_{t_{0}}\).

Thus, condition (25) of Theorem 3.2 does not hold. In fact, we can easily verify that \(x(t)=t^{(\frac{1}{3})}\) is a nonoscillatory solution of Eq. (36).

Declarations

Acknowledgements

This work is supported by the National Natural Science Foundation of China (10971018). The author thanks the referees very much for their valuable comments and suggestions on this paper.

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